IGCSE Additional Mathematics: Logarithmic and Exponential Functions Practice Questions
A logarithm answers the question what power gives this number. The three laws convert products into sums, quotients into differences, and powers into multipliers, which is what makes exponential equations solvable.
Topic 7 of Cambridge IGCSE Additional Mathematics 0606 combines the laws of logarithms with equation solving. The hidden quadratic in an exponential equation is the highest scoring question type. The questions below build to it.
What you need to know for Logarithmic and Exponential Functions
- DefinitionIf a to the power x equals b, then the logarithm of b to base a equals x. Logarithms and exponentials are inverse operations.
- The three lawsThe log of a product is the sum of the logs. The log of a quotient is the difference. The log of a power brings the index down as a multiplier.
- Solving with logsTake logarithms of both sides, bring the index down, then divide.
- Hidden quadraticAn equation containing a to the power 2x and a to the power x becomes a quadratic under the substitution y equals a to the power x.
- Domain restrictionThe logarithm of a negative number or of zero is undefined, so any solution making an argument non-positive must be rejected.
IGCSE Additional Mathematics Logarithmic and Exponential Functions questions and answers
4 exam-style questions written to the 0606 syllabus. Try each one on paper first, then open the worked answer to check your method against the marks.
Solve the equation log3(x) + log3(x minus 2) = 1.
Show the worked answer
- Combine the left side using the product law: log3[x(x minus 2)] = 1.
- Convert to index form: x(x minus 2) = 31 = 3.
- Rearrange: x2 minus 2x minus 3 = 0, which factorises to (x minus 3)(x + 1) = 0.
- So x = 3 or x = minus 1. Reject x = minus 1, because the logarithm of a negative number is undefined. The solution is x = 3.
Solve 5x = 20, giving your answer correct to 3 decimal places.
Show the worked answer
- Take logarithms of both sides: x log 5 = log 20.
- Divide: x = log 20 divided by log 5.
- x = 1.30103 divided by 0.69897 = 1.8614, which is 1.861 to 3 decimal places.
Express log(a2b divided by the square root of c) in terms of log a, log b and log c.
Show the worked answer
- Apply the quotient law: log(a2b) minus log(the square root of c).
- Apply the product law to the first term: log(a2) + log b.
- Apply the power law: log(a2) = 2 log a, and the square root of c is c to the power one half, so its log is one half log c.
- Combining gives 2 log a + log b minus one half log c.
Solve 22x minus 5(2x) + 4 = 0.
Show the worked answer
- Note that 22x equals (2x)2, so substitute y = 2x.
- The equation becomes y2 minus 5y + 4 = 0.
- Factorise: (y minus 1)(y minus 4) = 0, so y = 1 or y = 4.
- Return to x. If 2x = 1 then x = 0. If 2x = 4 then x = 2.
- Both are valid, since 2 to any power is positive and both values of y are positive.
Common mistakes in this topic
- Treating the log of a sum as the sum of the logs.
- Failing to reject solutions that make a logarithm undefined.
- Dividing two logarithms and simplifying as though it were a quotient law.
- Forgetting to reverse the substitution in a hidden quadratic.
- Losing the sign on a term from the denominator.
Exam tips
- There is no law for the log of a sum. If you find yourself splitting one, stop.
- After solving a logarithmic equation, always check each root makes every argument positive.
- Spot a hidden quadratic when you see both a to the power 2x and a to the power x.
- When the bases cannot be matched, take logs of both sides.
- State the rejected root and why. The reason carries the mark.
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Logarithmic and Exponential Functions FAQs
What are the laws of logarithms?
The logarithm of a product equals the sum of the logarithms. The logarithm of a quotient equals the difference. The logarithm of a power brings the index down as a multiplier. There is no law for the logarithm of a sum.
How do I solve an exponential equation?
If both sides can be written to the same base, equate the indices. If not, take logarithms of both sides, use the power law to bring the index down, then divide to isolate the unknown.
Why must I reject some solutions of a logarithmic equation?
A logarithm is only defined for a positive argument. Combining logarithms and solving can produce a value that makes one of the original arguments zero or negative, which is not a valid solution and must be rejected with a stated reason.
How do I recognise a hidden quadratic?
Look for a term in a to the power 2x alongside a term in a to the power x. Substituting y equals a to the power x turns the equation into a standard quadratic. After solving for y, remember to convert each value back to find x.
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Written to the published Cambridge IGCSE Additional Mathematics (0606) syllabus. Check your school entry code and syllabus year, because Core and Extended candidates are assessed on different content. Last reviewed 2026-08-12.